 \documentclass[12pt]{article}
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\usepackage{epstopdf}
\graphicspath{{./Graphs/}}
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\newtheorem{definition}{Definition}
\newtheorem{claim}{Claim}
\newtheorem{example}{Example}
\newtheorem{proposition}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{remark}{Remark}
\newtheorem{corollary}{Corollary}
\newtheorem{implication}{Implication}
\newtheorem{assumption}{Assumption}
\newtheorem{property}{Property}
\newcommand{\bit}{\begin{itemize}}


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\numberwithin{property}{section}
\usepackage{graphicx}
\usepackage{tikz}
\graphicspath{ {./Graphs/} }
\begin {document} 


\begin{figure}
\begin{center}
\begin{tikzpicture}
\begin{axis}[
    axis lines = middle,
    xlabel = $x_1$,
    ylabel = {$E[U_V(x_1)]$}, 
    y label style={at={(axis description cs:0.5,-0.1)},anchor=north},
    x label style={at={(axis description cs:1,1)},anchor=south},
    yticklabels={,,,,},
    xticklabels={,,,}
]
   
   \addplot[color=gray,dashed,thick,mark=*]
   coordinates {
         
              (-0.35,-1.94)
              (1.19,-1.94)
              (0.08,-1.94)
        };
        
        \node[below left] at (axis cs: 1.15,-1.94){$\bar{x}$};
  \node[below right] at (axis cs: 0.08,-1.94){$\underline{x}$};
  \node[above] at (axis cs: -0.35,-1.94){$x^{neg}_V$};
   \addplot[color=black,solid,mark=*]
   coordinates {
         
              (17/25,0)
              (-1,0)
              (1,0)
              
             
        };
       
   \node[below] at (axis cs: 17/25,0) {$x^m_V$};
       
   \node[below] at (axis cs: 1,0) {$x'$};      
          
   \node[below] at (axis cs: -1,0) {$-x'$}; 
   \addplot [
    domain=-1.25:-1, 
    samples=100, 
    color=black,
    style=thick,
]
{-3/5*(x-1)^2 -2/5*(x+1)^2};
\addplot [
    domain=-1:0, 
    samples=100, 
    color=black,
    style=thick,
]
{-3/5*(x-1)^2 -2/5*(x+1)^2-(1+x)*(130/125)};

\addplot [
    domain=1:1.25, 
    samples=100, 
    color=black,
    style=thick,
]
{-3/5*(x-1)^2 -2/5*(x+1)^2};

\addplot [
    domain=0:1,
    samples=100, 
    color=black,
    style=thick,
]
{-3/5*(x-1)^2 -2/5*(x+1)^2-(1-x)*(130/125)};
\end{axis}
\end{tikzpicture}
\caption{Forward looking voter's expected utility as a function of first-period policy}
\end{center}
\end{figure}
\end{document} 